Harris scored 17 of his points in the second half as the Great Danes (3-3) held off a late run.
Velton Jones, who scored 17 points, helped Robert Morris (1-4), the defending Northeast Conference champion, hit consecutive jumpers to cut Albany's lead to 67-63.
Albany's Derrek Tartt hit two free throws with 16.4 seconds left and Harris added two more in the final seconds to seal the victory.
Albany led by 11 points with 4:14 remaining.
The Great Danes, who shot 58.3 percent in the first half, led 35-27 at halftime after Robert Morris shot only 30 percent from the field.
Mezie Nwigwe led Robert Morris with 18 points.